Theorem bd0 17016

Description: A formula equivalent to a bounded one is bounded. See also bd0r 17017. (Contributed by BJ, 3-Oct-2019.)

Hypotheses

Ref	Expression
bd0.min	⊢ BOUNDED 𝜑
bd0.maj	⊢ (𝜑 ↔ 𝜓)

Assertion

Ref	Expression
bd0	⊢ BOUNDED 𝜓

Proof of Theorem bd0

Step	Hyp	Ref	Expression
1		bd0.min	. 2 ⊢ BOUNDED 𝜑
2		bd0.maj	. . 3 ⊢ (𝜑 ↔ 𝜓)
3	2	ax-bd0 17005	. 2 ⊢ (BOUNDED 𝜑 → BOUNDED 𝜓)
4	1, 3	ax-mp 5	1 ⊢ BOUNDED 𝜓

This proof depends on syntax axioms:   ↔ wb 105  BOUNDED wbd 17004

This proof depends on axioms:  ax-mp 5  ax-bd0 17005

This theorem is used by:  bd0r  17017  bdth  17023  bdnth  17026  bdnthALT  17027  bdph  17042  bdsbc  17050  bdsnss  17065  bdcint  17069  bdeqsuc  17073  bdcriota  17075  bj-axun2  17107